Uncertain Chad asks “What’s your favorite dubious proof technique?” I just don’t have one dubious proof technique: I have an entire book of dubious proof techniques! Seriously, I have a book where I write them all down.
But if I had to choose a dubious proof technique that was my favorite, it would have to be “proof by logical exhaustion.” Now you might think that this means that I logically list all possibilities to prove something, a technique which is perfectly valid and not very dubious at all. No, no. “Proof by logical exhaustion” is where you put forth a chain of logical reasoning so long and so tedious that your opponent is sure to get lost in the argument and in exhaustion will concede defeat.
I like this proof technique because it reminds me of a manuscript I saw once claiming to disprove special relativity which used about seven or eight changes of reference frame. I mean the argument was wrong, but showing this left you pretty near brain dead from following the reasoning (and finding the hole in the reasoning.)